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Let $R$ be a commutative unital associative ring and set $R<x,y>$ to be the $R$-algebra of non-commuting polynomials in two variables over $R$.

Explicitly how would one go about computing exactly what the module of $R$-differentials from $R<x,y>$ to itself?

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The module of $R$-derivations is just $R<x,y>^2$: given $P,Q$ in $R<x,y>$, there is one and only one derivation $D$ such that $Dx=P$ and $Dy=Q$. See Bourbaki, Algebra III, §10, Proposition 14. – abx Apr 11 '14 at 5:56
Perfect, I'll look it up and post the answer to this question, thanks ABx – CSA Apr 11 '14 at 18:25
This is not really a question appropriate for this site. The free álgebra has a universal property with respect to derivations which makes the answer obvious. – Mariano Suárez-Alvarez Apr 18 '14 at 19:47

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